lecture22 - Generally speaking, a differential equation is...

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§ 4.3 Differentiation of Exponential Functions Math 121 Lecture 22
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Chain Rule for e g ( x ) Working With Differential Equations Solving Differential Equations at Initial Values Functions of the form e kx
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d dx e kx = e kx d dx kx ( ) = ke kx
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Differentiate. f x ( ) = e x + 5 This is the given function. Use the chain rule. f x ( ) = e x + 5 d dx x + 5 ( ) Simplify. = e x + 5 f x ( ) = e x + 5
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Differentiate. g x ( ) = e 2 x 2 x ( ) 3 This is the given function. g x ( ) = e 2 x 2 x ( ) 3 Use the chain rule. g x ( ) = 3 e 2 x 2 x ( ) 2 d dx e 2 x 2 x ( ) Remove parentheses. g x ( ) = 3 e 2 x 2 x ( ) 2 d dx e 2 x d dx 2 x Use the chain rule for exponential functions. g x ( ) = 3 e 2 x 2 x ( ) 2 2 e 2 x 2 ( )
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Unformatted text preview: Generally speaking, a differential equation is an equation that contains a derivative. Determine all solutions of the differential equation y = 1 2 y . The equation has the form y = ky with k = 1/2. Therefore, any solution of the equation has the form y = 1 2 y y = Ce 1 2 x where C is a constant. Determine all functions y = f ( x ) such that y = 3 y and f (0) = . The equation has the form y = ky with k = 3. Therefore, for some constant C . We also require that f (0) = . That is, So C = and...
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lecture22 - Generally speaking, a differential equation is...

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